16 BAIRE'S THEOREM AND ITS CONSEQUENCES 87 coefficients cn e E' are given by Applying this result to the situation where 7 is a circuit (Q^t<>2n) in A - {a} we obtain, for any weakly analytic function on A, the Taylor expansion (12.16.6.5) /,(*) = £ 1 /<»>(*)(Z - a)", n=0 Wl convergent in the disk \z — a\ < r (independent ofze E), and the derivatives are given by More generally, suppose that in A — {a} we have (12.16.6.7) /z=^_^i_ + ^> where the x% are linear forms on E (a priori, not necessarily continuous), and z\~*gz is a weakly analytic function on A — {a} which is weakly bounded in a neighborhood of a. Then, by (9.15), for each x e E, the function z\-+gs(x) can be extended by continuity to the point a, and the extended function is analytic in A. Hence there exists a weakly analytic function zt-*hz in A whose restric- tion to A - {a} is z\-*gz. Furthermore, the x* are continuous linear forms on E (i.e., they are elements of E'). Indeed, for each x e E we have = ^ J(z - a)* where y is a circuit t\~+a + reu (0 <J tf <i 27c) with r sufficiently small and independent of x (9.14); our assertion therefore follows from the considera- tions above (12.16.6.3). Finally, we remark that the principle of analytic continuation (9.4.2) remains valid for weakly analytic functions. This is an immediate consequence of the definitions. (12.16.7) Let E be a Hilbert space and u an endomorphism of the (non- topologicaT) vector space E. Then the following three conditions are equivalent: (a) u is continuous] (b) u is weakly continuous•; (c) u has an adjoint (11.5).oints (zn) in A (3.13.13),